By Lm. REF , we can choose the Stinespring dilation \(\phi _\rho = v^* \Phi _\rho v\)  to be of the following simple form: \(v: \mathcal {H}_\mathcal {B}\rightarrow \mathcal {K}\)  for \(\mathcal {K}= \mathcal {H}_\mathcal {B}\otimes \mathcal {H}_\mathcal {A}\)For a general Stinespring dilation, one needs \(\mathcal {K}= \mathcal {H}_\mathcal {B}\otimes \mathcal {H}_\mathcal {A}\otimes \mathcal {H}_\mathcal {A}\)  (cf. [1]}, [2]}). However, as it turns out in the case that both \(\Phi \)  and \(\Phi ^*\)  are both \(C^*\) -homomorphisms, \(\mathcal {K}\)  can be reduced by one factor of \(\mathcal {H}_\mathcal {A}\) .
and \(\Phi _\rho = 1_\mathcal {B}\otimes \mathrm {id}_{\mathrm {supp}(\Phi _\rho )}\) .This representation allows to interpret a quantum channel \(\phi \)  as a coarse-grained unitary bipartite channel on the target and some unknown ancillary system, after tracing out the latter. In particular, generalised measurements can be understood as projective measurements on a larger system [3]}.
In fact, if \(\Phi ^*_\rho \)  is a \(C^*\) -homomorphism, then \(\phi _{\rho ^{T_\mathcal {A}}} = \phi ^*_\rho = v^*\Phi ^*_\rho v\)  is a Stinespring dilation (see Lm. REF ).In particular, \(\rho ^{T_\mathcal {A}}\)  is positive in this case. By Lm. REF , \(\Phi _\rho (\mathcal {A}) \subset \mathcal {B}(\mathcal {K})\)  is a commutative subalgebra
and there exists a commutative subalgebra \(V \subset \mathcal {A}\)  such that \(\Phi _\rho (V) = \Phi _\rho (\mathcal {A}) \subset \mathcal {B}(\mathcal {K})\) . We thus have \(\Phi _\rho (a) = \Phi _\rho |_V(a) = 1_\mathcal {B}\otimes p_Vap_V\) , where \(p_V\)  denotes the projection onto \(V\) .
